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Network EvolutionStatistics of Networks
Comparing Networks
Networks in Cellular Biology
A. Metabolic Pathways
B. Regulatory Networks
C. Signaling Pathways
D. Protein Interaction Networks - PIN
Empirical Facts
Dynamics on Networks (models)
Models of Network Evolution
Network Alignment & MotifsBarabasi & Oltvai, 2004, Sharan & Ideker, 2006
•Motifs
•Network Alignment
1. Are nodes/edges labelled?2. Which operations are allowed?3. Pair/Multiple?
•Network Search
Find (approximately) a network within a set of others.
•Network integration
Combine a set of networks to one large network.
Network Description and Statistics IBarabasi & Oltvai, 2004
Remade from Barabasi, 2004
•Degree Distribution - P(k)
•Scale Free Networks P(k)~k-
•Hubs: multiply connected nodes
The lower , the more hubs.
Small World Property:
Graph connected and path lengths small
•Degree/Indegree/Outdegree
•Shortest Path
•Mean Path Length
•Diameter:
•Clustering Coefficient - CI=2TI/nI(nI-1)
CA=1/10
Maxi, j{Dist(i, j)}
Dist(i, j)
Network Description and Statistics IIBarabasi & Oltvai, 2004
C.Hierarchial
Copy smaller graphs and let them keep their connections.
Phase transition:
Connected if:
A. Random Networks [Erdos and Rényi (1959, 1960)]
Mean path length ~ ln(k)
B. Scale Free [Price,1965 & Barabasi,1999]
Preferential attachment. Add proportionally to connectedness
Mean path length ~ lnln(k)
A. Metabolic Pathways
S P
I2
I4
I3
I1
•Flux Analysis
•Metabolic Control Theory
•Biochemical Systems Theory
•Kinetic Modeling
Control Coefficients(Heinrich & Schuster: Regulation of Cellular Systems. 1996)
Flux Control Coeffecient – FCC:
)ln(
)ln()( 0
k
j
k
j
j
kE
k
j
j
kJE E
J
E
J
J
E
E
J
J
EC
k
j
k
Kacser & Burns, 73
)ln(
)ln()( 0
k
j
k
j
j
kv
k
j
j
kJ JJ
J
J
JC
k
j
k
Heinrich & Rapoport, 73-74
Flux Jj (edges) – Enzyme conc., Ek (edges), S – internal nodes.
P1 S1 P2
E,
FCF: gluconeogenesis from lactate
Pyruvate transport .01
Pyruvate carboxylase .83
Oxaloacetate transport .04
PEOCK .08
A Model for Network Inference
•A core metabolism:
•A given set of metabolites:
•A given set of possible reactions -
arrows not shown.
•A set of present reactions - M
black and red arrows
Let be the rate of deletion the rate of insertionThen
Restriction R:
A metabolism must define a connected graph
M + R defines
1. a set of deletable (dashed) edges D(M):
2. and a set of addable edges A(M):
dP(M)
dt P(M ') P(M ' ')
M ''A (M )
M 'D(M )
- P(M)[ D(M) A(M) ]
Likelihood of Homologous PathwaysNumber of Metabolisms:
1 2
3 4
+ 2 symmetrical versions
P( , )=P( )P( -> )
Eleni Giannoulatou
Approaches: Continuous Time Markov Chains with computational tricks.
MCMC
Importance Sampling
Remade from Somogyi & Sniegoski,96. F2
A B
A BmRNA mRNAFactor A Factor B
A B
A B
C
C
A B
A B
C
C
mRNA
mRNA
Factor C
Factor B
mRNAFactor A
mRNA
mRNA
Factor C
Factor B
mRNAFactor A
B. Regulatory Networks
Remade from Somogyi & Sniegoski,96. F4
Boolen functions, Wiring Diagrams and Trajectories
A B
A B
C
C
Inputs 2 1 1Rule 4 2 2
A activates B
B activates C
A is activated by B, inhibited by (B>C)
Point Attractor
A B C1 1 01 1 10 1 10 0 10 0 00 0 0
2 State Attractor
A B C1 0 00 1 01 0 10 1 0
Contradiction: Always turned off (biological meaningless) Tautology: Always turned on (household genes)
k=1:
input
output
0
10 or 1
4k=2:
input
output
0,0
0,1
1,0
1,1
0 or 1
16
For each gene dependent on i genes: genes.dependent of choices
i
kki
i
k)2 ( Rules BooleanofNumber
A single function:
k2kk2
The whole set:
Gene 2
Gene n
Gene 1
Time 1 Time 2 Time 3 Time T
Boolean NetworksR.Somogyi & CA Sniegoski (1996) Modelling the Complexity of Genetic Networks Complexity 1.6.45-64.
Reverse Engeneering Algorithm-RevealD’haeseler et al.(2000) Genetic network Inference: from co-expression clustering to reverse engineering. Bioinformatics 16.8.707-
Assumptions:
Discrete known Generations
No Noise
0 1 1 1 1 0
0 0 1 1 0 1
1 1 1 0 1 0
BOOL-1Akutsu et al. (2000) Inferring qualitative relations in genetic networks and metabolic pathways. Bioinformatics 16.2.727-
If O(22k[2k + ]log(n)) INPUT patterns are given uniformly randomly, BOOL-1 correctly
identifies the underlying network with probability 1-n, where is any fixed real number > 1.
Algorithm
For each gene do (n)
For each boolean rule (<= k inputs) not violated, keep it.
C. Signaling Pathways
www.hprd.org from Pierre deMeyts
•Transmits signals from membrane to gene regulation.
•Its function is enigmatic as some of the molecules involved are common to different functions and how cross-interaction is avoided is unknown.
D. Protein Interaction Network
Yea
st p
rote
in in
tera
ctio
n n
etw
ork
[Je
on
g e
t al
., N
atu
re (
20
01
)]•The sticking together of different protein is measured by mass spectroscopy.
•The nodes will be all known proteins.
•Two nodes are connected if they stick together. This can be indicator of being part of a a functional protein complex, but can also occur for other reasons.
PIN Network EvolutionBarabasi & Oltvai, 2004 & Berg et al. ,2004; Wiuf etal., 2006
•A gene duplicates
•Inherits it connections
•The connections can change
Berg et al. ,2004:
•Gene duplication slow ~10-9/year
•Connection evolution fast ~10-6/year
•Observed networks can be modeled as if node number was fixed.
Likelihood of PINsIrreducible (and isomorphic)
Data
de-DAing
De-con
nectin
g
2386 nodes and 7221 links
735 nodes
)0,33,.66,.1(0
•Can only handle 1 graph.
•Limited Evolution Model
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